On the Extra Mode and Inconsistency of Horava Gravity

Abstract: We address the consistency of Horava's proposal for a theory of quantum gravity from the low-energy perspective. We uncover the additional scalar degree of freedom arising from the explicit breaking of the general covariance and study its properties. The analysis is performed both in the original formulation of the theory and in the Stueckelberg picture. A peculiarity of the new mode is that it satisfies an equation of motion that is of first order in time derivatives. At linear level the mode is manifest only around spatially inhomogeneous and time-dependent backgrounds. We find two serious problems associated with this mode. First, the mode develops very fast exponential instabilities at short distances. Second, it becomes strongly coupled at an extremely low cutoff scale. We also discuss the "projectable" version of Horava's proposal and argue that this version can be understood as a certain limit of the ghost condensate model. The theory is still problematic since the additional field generically forms caustics and, again, has a very low strong coupling scale. We clarify some subtleties that arise in the application of the Stueckelberg formalism to Horava's model due to its non-relativistic nature.

• The paper unveils an extra scalar degree of freedom caused by the foliation structure, resulting in a first-order time derivative equation.
• It identifies exponential instabilities and a low energy strong coupling scale that undermine the theory's ultraviolet behavior.
• The analysis contrasts projectable and non-projectable models, highlighting challenges in recovering General Relativity at low energies.

Analysis of the Extra Mode and Consistency Challenges in Hořava Gravity

The paper addresses the consistency and dynamics of Hořava gravity, a theory proposed to offer a renormalizable account of quantum gravity by modifying spacetime structure through the introduction of a foliation by spacelike surfaces, thereby explicitly breaking general covariance inherent in General Relativity (GR). The theory intends to improve the ultraviolet (UV) behavior of the graviton propagator.

Key Findings and Theoretical Implications

1. Extra Scalar Degree of Freedom: The study unveils an additional scalar degree of freedom, which appears due to the breaking of general covariance from the foliation structure incorporated in Hořava gravity. Notably, this mode exhibits an atypical equation of motion, being first order in time derivatives, a peculiarity that distinguishes it from standard scalar modes in most gravitational theories.
2. Instabilities and Strong Coupling Issues: The paper presents two significant issues stemming from the extra mode:
   • Exponential Instabilities: This mode demonstrates rapid exponential instabilities, specifically at short distances, which poses challenges to the fundamental stability of the theory.
   • Low Cutoff Strong Coupling Scale: The mode becomes strongly coupled at an extremely low energy scale, which is insufficient for the energy range anticipated in quantum gravity applications. This observation suggests that the strong coupling renders the theory reliable only within a restricted window of small energies, far below the Planck scale.
3. Projectable vs. Non-Projectable Versions: The analysis differentiates between the "projectable" and "non-projectable" versions of Hořava gravity. The projectable version results in dynamics resembling ghost condensation models and reduces to a dust-like behavior but faces issues with caustics and retains a low strong coupling threshold, complicating the practical integration into a quantum gravity framework.

Theoretical and Practical Implications

• Challenges in GR Recovery: The breakdowns identified in the non-projectable model highlight the difficulty in recovering GR as a low-energy limit of Hořava gravity. The introduction of extra degrees of freedom complicates this continuity, with instabilities and strong coupling presenting barriers to achieving a consistent infrared (IR) behavior akin to GR.
• Reduction of Degrees of Freedom: A central theoretical contribution is the identification of scenarios where higher-derivative operators in covariant formulations could potentially lead to a reduction in degrees of freedom in the preferred foliation. This insight could influence future model building, where similar mechanisms are employed to control the propagation of additional modes.
• Future Directions in Quantum Gravity: While the paper highlights substantial limitations in the current form of Hořava gravity, the insights into handling extra modes and constraints might inform alternative approaches to modify gravity at quantum scales. Models utilizing similar symmetry-breaking techniques may derive from this work, though much progress is required to ensure stable and weakly-coupled dynamics.

Overall, the paper distinctly analyzes the breakdowns in Hořava gravity due to extra modes, contributing to a critical understanding of alternative formulations of gravity with explicit Lorentz violation. The findings urge caution and further scrutiny in developing quantum gravity theories with similar structural assumptions. Future developments may heed the issues raised, seeking ways to resolve inherent instabilities and ensure a robust theoretical framework.